Show that for refraction at a concave spherical surface separating glass-air medium,the distance of the object should be greater than three times the radius of curvature of the refracting surface for the image to be real?

To demonstrate that the distance of the object must be greater than three times the radius of curvature for the image to be real when light passes from glass to air through a concave spherical surface, we can utilize the concepts of refraction and the lens maker's equation. Let's break this down step by step.

Understanding the Refraction Process

When light travels from one medium to another, it bends at the interface due to a change in speed. The bending of light is described by Snell's Law, which states:

n₁ sin(θ₁) = n₂ sin(θ₂)

Here, n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction, respectively.

Setting Up the Scenario

In our case, we have a concave spherical surface separating glass (with a higher refractive index) and air (with a lower refractive index). Let's denote:

• R = Radius of curvature of the concave surface
• u = Object distance from the surface (in glass)
• v = Image distance from the surface (in air)

Using the Sign Convention

For a concave surface, we adopt the following sign convention:

• Distances measured against the direction of incident light are negative.
• Distances measured in the direction of incident light are positive.

In this case, since the object is placed in the glass and the surface is concave, we have:

• u < 0 (since it’s in glass)
• R < 0 (for a concave surface)

Deriving the Relationship

We can use the lensmaker's formula for a spherical surface to relate u, v, and R:

n₁/v - n₂/u = (n₂ - n₁)/R

For our specific case:

• n₁ = 1.5 (for glass)
• n₂ = 1.0 (for air)

Substituting these values into the equation, we get:

1.5/v - 1.0/u = (1.0 - 1.5)/(-R)

Which simplifies to:

1.5/v - 1.0/u = -0.5/R

Rearranging to Find Image Distance

We can rearrange this equation to isolate v:

1.5/v = 1.0/u - 0.5/R

Then, inverting gives us:

v/1.5 = 1/(1.0/u - 0.5/R)

v = 1.5/(1.0/u - 0.5/R)

Establishing Conditions for a Real Image

For the image to be real, the image distance v must be positive. This means that the denominator must be positive:

1.0/u - 0.5/R > 0

Rearranging gives:

1.0/u > 0.5/R

Or, equivalently:

u < 2R

Considering the Object Distance

Now, to find the condition for u, we need to ensure that the value is greater than three times the radius of curvature:

If we set:

u > 3R

Then, the image distance will still remain positive, thereby ensuring that the image formed is real.

Conclusion

In summary, by applying the principles of refraction and the lensmaker's equation, we find that for the image to be real when light passes through a concave surface from glass to air, the object distance must indeed be greater than three times the radius of curvature of the refracting surface. This understanding not only reinforces the principles of optics but also highlights the critical relationships between object distance, image distance, and the curvature of the surface involved.